LMFDB - Embedded newform 2254.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.990305153$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 38x^{4} + 203x^{2} - 100 $ x^6 - 38x^4 + 203x^2 - 100
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.740283$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -5.32887 q^{3} +4.00000 q^{4} +6.69892 q^{5} -10.6577 q^{6} +8.00000 q^{8} +1.39685 q^{9} +13.3978 q^{10} -9.88974 q^{11} -21.3155 q^{12} +13.2146 q^{13} -35.6977 q^{15} +16.0000 q^{16} -73.3216 q^{17} +2.79370 q^{18} +29.4542 q^{19} +26.7957 q^{20} -19.7795 q^{22} +23.0000 q^{23} -42.6310 q^{24} -80.1245 q^{25} +26.4291 q^{26} +136.436 q^{27} +239.701 q^{29} -71.3953 q^{30} +204.202 q^{31} +32.0000 q^{32} +52.7011 q^{33} -146.643 q^{34} +5.58741 q^{36} -416.948 q^{37} +58.9083 q^{38} -70.4187 q^{39} +53.5914 q^{40} +372.338 q^{41} -366.186 q^{43} -39.5590 q^{44} +9.35739 q^{45} +46.0000 q^{46} -306.824 q^{47} -85.2619 q^{48} -160.249 q^{50} +390.721 q^{51} +52.8583 q^{52} +282.051 q^{53} +272.872 q^{54} -66.2506 q^{55} -156.957 q^{57} +479.402 q^{58} -393.159 q^{59} -142.791 q^{60} -938.136 q^{61} +408.403 q^{62} +64.0000 q^{64} +88.5233 q^{65} +105.402 q^{66} -424.530 q^{67} -293.286 q^{68} -122.564 q^{69} -388.680 q^{71} +11.1748 q^{72} +579.008 q^{73} -833.896 q^{74} +426.973 q^{75} +117.817 q^{76} -140.837 q^{78} -946.393 q^{79} +107.183 q^{80} -764.764 q^{81} +744.676 q^{82} +1420.98 q^{83} -491.175 q^{85} -732.372 q^{86} -1277.33 q^{87} -79.1179 q^{88} +490.678 q^{89} +18.7148 q^{90} +92.0000 q^{92} -1088.16 q^{93} -613.648 q^{94} +197.311 q^{95} -170.524 q^{96} +1108.65 q^{97} -13.8145 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 24 q^{4} + 48 q^{8} - 52 q^{9} - 84 q^{11} + 52 q^{15} + 96 q^{16} - 104 q^{18} - 168 q^{22} + 138 q^{23} - 286 q^{25} - 22 q^{29} + 104 q^{30} + 192 q^{32} - 208 q^{36} - 180 q^{37} - 414 q^{39}+ \cdots + 600 q^{99}+O(q^{100}) $ 6 * q + 12 * q^2 + 24 * q^4 + 48 * q^8 - 52 * q^9 - 84 * q^11 + 52 * q^15 + 96 * q^16 - 104 * q^18 - 168 * q^22 + 138 * q^23 - 286 * q^25 - 22 * q^29 + 104 * q^30 + 192 * q^32 - 208 * q^36 - 180 * q^37 - 414 * q^39 - 600 * q^43 - 336 * q^44 + 276 * q^46 - 572 * q^50 + 464 * q^51 - 876 * q^53 - 1452 * q^57 - 44 * q^58 + 208 * q^60 + 384 * q^64 - 1532 * q^65 - 92 * q^67 - 1434 * q^71 - 416 * q^72 - 360 * q^74 - 828 * q^78 - 2820 * q^79 - 2198 * q^81 - 184 * q^85 - 1200 * q^86 - 672 * q^88 + 552 * q^92 - 2742 * q^93 - 2888 * q^95 + 600 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−5.32887	−1.02554	−0.512771	−	0.858526i	$-0.671381\pi$
$3$	−5.32887	−1.02554	−0.512771	+	0.858526i	$0.671381\pi$
$4$	4.00000	0.500000
$5$	6.69892	0.599170	0.299585	−	0.954070i	$-0.403152\pi$
$5$	6.69892	0.599170	0.299585	+	0.954070i	$0.403152\pi$
$6$	−10.6577	−0.725167
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	1.39685	0.0517352
$10$	13.3978	0.423677
$11$	−9.88974	−0.271079	−0.135539	−	0.990772i	$-0.543277\pi$
$11$	−9.88974	−0.271079	−0.135539	+	0.990772i	$0.543277\pi$
$12$	−21.3155	−0.512771
$13$	13.2146	0.281928	0.140964	−	0.990015i	$-0.454980\pi$
$13$	13.2146	0.281928	0.140964	+	0.990015i	$0.454980\pi$
$14$	0	0
$15$	−35.6977	−0.614473
$16$	16.0000	0.250000
$17$	−73.3216	−1.04606	−0.523032	−	0.852313i	$-0.675199\pi$
$17$	−73.3216	−1.04606	−0.523032	+	0.852313i	$0.675199\pi$
$18$	2.79370	0.0365823
$19$	29.4542	0.355645	0.177822	−	0.984063i	$-0.443095\pi$
$19$	29.4542	0.355645	0.177822	+	0.984063i	$0.443095\pi$
$20$	26.7957	0.299585
$21$	0	0
$22$	−19.7795	−0.191682
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−42.6310	−0.362584
$25$	−80.1245	−0.640996
$26$	26.4291	0.199353
$27$	136.436	0.972485
$28$	0	0
$29$	239.701	1.53487	0.767436	−	0.641125i	$-0.221532\pi$
$29$	239.701	1.53487	0.767436	+	0.641125i	$0.221532\pi$
$30$	−71.3953	−0.434498
$31$	204.202	1.18309	0.591544	−	0.806273i	$-0.298519\pi$
$31$	204.202	1.18309	0.591544	+	0.806273i	$0.298519\pi$
$32$	32.0000	0.176777
$33$	52.7011	0.278003
$34$	−146.643	−0.739679
$35$	0	0
$36$	5.58741	0.0258676
$37$	−416.948	−1.85259	−0.926295	−	0.376798i	$-0.877025\pi$
$37$	−416.948	−1.85259	−0.926295	+	0.376798i	$0.877025\pi$
$38$	58.9083	0.251479
$39$	−70.4187	−0.289129
$40$	53.5914	0.211838
$41$	372.338	1.41828	0.709139	−	0.705069i	$-0.249084\pi$
$41$	372.338	1.41828	0.709139	+	0.705069i	$0.249084\pi$
$42$	0	0
$43$	−366.186	−1.29867	−0.649335	−	0.760502i	$-0.724953\pi$
$43$	−366.186	−1.29867	−0.649335	+	0.760502i	$0.724953\pi$
$44$	−39.5590	−0.135539
$45$	9.35739	0.0309982
$46$	46.0000	0.147442
$47$	−306.824	−0.952231	−0.476115	−	0.879383i	$-0.657956\pi$
$47$	−306.824	−0.952231	−0.476115	+	0.879383i	$0.657956\pi$
$48$	−85.2619	−0.256385
$49$	0	0
$50$	−160.249	−0.453253
$51$	390.721	1.07278
$52$	52.8583	0.140964
$53$	282.051	0.730994	0.365497	−	0.930813i	$-0.380899\pi$
$53$	282.051	0.730994	0.365497	+	0.930813i	$0.380899\pi$
$54$	272.872	0.687651
$55$	−66.2506	−0.162422
$56$	0	0
$57$	−156.957	−0.364728
$58$	479.402	1.08532
$59$	−393.159	−0.867542	−0.433771	−	0.901023i	$-0.642817\pi$
$59$	−393.159	−0.867542	−0.433771	+	0.901023i	$0.642817\pi$
$60$	−142.791	−0.307237
$61$	−938.136	−1.96911	−0.984557	−	0.175064i	$-0.943987\pi$
$61$	−938.136	−1.96911	−0.984557	+	0.175064i	$0.943987\pi$
$62$	408.403	0.836569
$63$	0	0
$64$	64.0000	0.125000
$65$	88.5233	0.168923
$66$	105.402	0.196578
$67$	−424.530	−0.774098	−0.387049	−	0.922059i	$-0.626506\pi$
$67$	−424.530	−0.774098	−0.387049	+	0.922059i	$0.626506\pi$
$68$	−293.286	−0.523032
$69$	−122.564	−0.213840
$70$	0	0
$71$	−388.680	−0.649688	−0.324844	−	0.945768i	$-0.605312\pi$
$71$	−388.680	−0.649688	−0.324844	+	0.945768i	$0.605312\pi$
$72$	11.1748	0.0182912
$73$	579.008	0.928325	0.464162	−	0.885750i	$-0.346355\pi$
$73$	579.008	0.928325	0.464162	+	0.885750i	$0.346355\pi$
$74$	−833.896	−1.30998
$75$	426.973	0.657368
$76$	117.817	0.177822
$77$	0	0
$78$	−140.837	−0.204445
$79$	−946.393	−1.34782	−0.673908	−	0.738815i	$-0.735386\pi$
$79$	−946.393	−1.34782	−0.673908	+	0.738815i	$0.735386\pi$
$80$	107.183	0.149792
$81$	−764.764	−1.04906
$82$	744.676	1.00287
$83$	1420.98	1.87919	0.939594	−	0.342292i	$-0.111203\pi$
$83$	1420.98	1.87919	0.939594	+	0.342292i	$0.111203\pi$
$84$	0	0
$85$	−491.175	−0.626770
$86$	−732.372	−0.918299
$87$	−1277.33	−1.57408
$88$	−79.1179	−0.0958409
$89$	490.678	0.584402	0.292201	−	0.956357i	$-0.405612\pi$
$89$	490.678	0.584402	0.292201	+	0.956357i	$0.405612\pi$
$90$	18.7148	0.0219190
$91$	0	0
$92$	92.0000	0.104257
$93$	−1088.16	−1.21331
$94$	−613.648	−0.673329
$95$	197.311	0.213091
$96$	−170.524	−0.181292
$97$	1108.65	1.16047	0.580237	−	0.814448i	$-0.302960\pi$
$97$	1108.65	1.16047	0.580237	+	0.814448i	$0.302960\pi$
$98$	0	0
$99$	−13.8145	−0.0140243
$100$	−320.498	−0.320498
$101$	−1138.84	−1.12196	−0.560982	−	0.827828i	$-0.689576\pi$
$101$	−1138.84	−1.12196	−0.560982	+	0.827828i	$0.689576\pi$
$102$	781.442	0.758572
$103$	395.385	0.378238	0.189119	−	0.981954i	$-0.439437\pi$
$103$	395.385	0.378238	0.189119	+	0.981954i	$0.439437\pi$
$104$	105.717	0.0996766
$105$	0	0
$106$	564.102	0.516891
$107$	1246.94	1.12660	0.563301	−	0.826252i	$-0.309531\pi$
$107$	1246.94	1.12660	0.563301	+	0.826252i	$0.309531\pi$
$108$	545.743	0.486242
$109$	437.697	0.384622	0.192311	−	0.981334i	$-0.438402\pi$
$109$	437.697	0.384622	0.192311	+	0.981334i	$0.438402\pi$
$110$	−132.501	−0.114850
$111$	2221.86	1.89991
$112$	0	0
$113$	1806.39	1.50381	0.751907	−	0.659269i	$-0.229134\pi$
$113$	1806.39	1.50381	0.751907	+	0.659269i	$0.229134\pi$
$114$	−313.915	−0.257902
$115$	154.075	0.124935
$116$	958.803	0.767436
$117$	18.4588	0.0145856
$118$	−786.318	−0.613445
$119$	0	0
$120$	−285.581	−0.217249
$121$	−1233.19	−0.926516
$122$	−1876.27	−1.39237
$123$	−1984.14	−1.45450
$124$	816.807	0.591544
$125$	−1374.11	−0.983235
$126$	0	0
$127$	−2481.90	−1.73412	−0.867059	−	0.498205i	$-0.833993\pi$
$127$	−2481.90	−1.73412	−0.867059	+	0.498205i	$0.833993\pi$
$128$	128.000	0.0883883
$129$	1951.36	1.33184
$130$	177.047	0.119446
$131$	1621.06	1.08117	0.540583	−	0.841291i	$-0.318204\pi$
$131$	1621.06	1.08117	0.540583	+	0.841291i	$0.318204\pi$
$132$	210.805	0.139001
$133$	0	0
$134$	−849.060	−0.547370
$135$	913.973	0.582683
$136$	−586.573	−0.369840
$137$	−659.952	−0.411559	−0.205779	−	0.978598i	$-0.565973\pi$
$137$	−659.952	−0.411559	−0.205779	+	0.978598i	$0.565973\pi$
$138$	−245.128	−0.151208
$139$	605.899	0.369724	0.184862	−	0.982764i	$-0.440816\pi$
$139$	605.899	0.369724	0.184862	+	0.982764i	$0.440816\pi$
$140$	0	0
$141$	1635.02	0.976552
$142$	−777.360	−0.459399
$143$	−130.689	−0.0764247
$144$	22.3496	0.0129338
$145$	1605.74	0.919649
$146$	1158.02	0.656425
$147$	0	0
$148$	−1667.79	−0.926295
$149$	−489.284	−0.269018	−0.134509	−	0.990912i	$-0.542946\pi$
$149$	−489.284	−0.269018	−0.134509	+	0.990912i	$0.542946\pi$
$150$	853.946	0.464829
$151$	−1947.38	−1.04950	−0.524752	−	0.851255i	$-0.675842\pi$
$151$	−1947.38	−1.04950	−0.524752	+	0.851255i	$0.675842\pi$
$152$	235.633	0.125739
$153$	−102.419	−0.0541184
$154$	0	0
$155$	1367.93	0.708870
$156$	−281.675	−0.144564
$157$	−1991.79	−1.01250	−0.506249	−	0.862388i	$-0.668968\pi$
$157$	−1991.79	−1.01250	−0.506249	+	0.862388i	$0.668968\pi$
$158$	−1892.79	−0.953050
$159$	−1503.01	−0.749664
$160$	214.365	0.105919
$161$	0	0
$162$	−1529.53	−0.741797
$163$	73.7887	0.0354575	0.0177288	−	0.999843i	$-0.494356\pi$
$163$	73.7887	0.0354575	0.0177288	+	0.999843i	$0.494356\pi$
$164$	1489.35	0.709139
$165$	353.041	0.166571
$166$	2841.95	1.32879
$167$	−1141.51	−0.528939	−0.264470	−	0.964394i	$-0.585197\pi$
$167$	−1141.51	−0.528939	−0.264470	+	0.964394i	$0.585197\pi$
$168$	0	0
$169$	−2022.38	−0.920517
$170$	−982.351	−0.443193
$171$	41.1431	0.0183994
$172$	−1464.74	−0.649335
$173$	−2772.24	−1.21832	−0.609160	−	0.793047i	$-0.708493\pi$
$173$	−2772.24	−1.21832	−0.609160	+	0.793047i	$0.708493\pi$
$174$	−2554.67	−1.11304
$175$	0	0
$176$	−158.236	−0.0677697
$177$	2095.09	0.889700
$178$	981.357	0.413235
$179$	−2613.36	−1.09124	−0.545618	−	0.838034i	$-0.683705\pi$
$179$	−2613.36	−1.09124	−0.545618	+	0.838034i	$0.683705\pi$
$180$	37.4296	0.0154991
$181$	−1272.00	−0.522361	−0.261180	−	0.965290i	$-0.584112\pi$
$181$	−1272.00	−0.522361	−0.261180	+	0.965290i	$0.584112\pi$
$182$	0	0
$183$	4999.20	2.01941
$184$	184.000	0.0737210
$185$	−2793.10	−1.11002
$186$	−2176.33	−0.857936
$187$	725.131	0.283566
$188$	−1227.30	−0.476115
$189$	0	0
$190$	394.622	0.150678
$191$	−2273.76	−0.861379	−0.430689	−	0.902500i	$-0.641730\pi$
$191$	−2273.76	−0.861379	−0.430689	+	0.902500i	$0.641730\pi$
$192$	−341.048	−0.128193
$193$	−3259.75	−1.21576	−0.607881	−	0.794028i	$-0.707980\pi$
$193$	−3259.75	−1.21576	−0.607881	+	0.794028i	$0.707980\pi$
$194$	2217.29	0.820579
$195$	−471.729	−0.173237
$196$	0	0
$197$	−5188.37	−1.87643	−0.938213	−	0.346058i	$-0.887520\pi$
$197$	−5188.37	−1.87643	−0.938213	+	0.346058i	$0.887520\pi$
$198$	−27.6290	−0.00991670
$199$	1457.14	0.519064	0.259532	−	0.965734i	$-0.416432\pi$
$199$	1457.14	0.519064	0.259532	+	0.965734i	$0.416432\pi$
$200$	−640.996	−0.226626
$201$	2262.26	0.793870
$202$	−2277.67	−0.793349
$203$	0	0
$204$	1562.88	0.536391
$205$	2494.26	0.849789
$206$	790.770	0.267454
$207$	32.1276	0.0107875
$208$	211.433	0.0704820
$209$	−291.294	−0.0964077
$210$	0	0
$211$	2010.75	0.656045	0.328022	−	0.944670i	$-0.393618\pi$
$211$	2010.75	0.656045	0.328022	+	0.944670i	$0.393618\pi$
$212$	1128.20	0.365497
$213$	2071.22	0.666282
$214$	2493.88	0.796628
$215$	−2453.05	−0.778124
$216$	1091.49	0.343825
$217$	0	0
$218$	875.395	0.271969
$219$	−3085.46	−0.952036
$220$	−265.002	−0.0812111
$221$	−968.913	−0.294915
$222$	4443.73	1.34344
$223$	721.559	0.216678	0.108339	−	0.994114i	$-0.465447\pi$
$223$	721.559	0.216678	0.108339	+	0.994114i	$0.465447\pi$
$224$	0	0
$225$	−111.922	−0.0331621
$226$	3612.78	1.06336
$227$	−2335.33	−0.682825	−0.341412	−	0.939914i	$-0.610905\pi$
$227$	−2335.33	−0.682825	−0.341412	+	0.939914i	$0.610905\pi$
$228$	−627.829	−0.182364
$229$	−125.168	−0.0361195	−0.0180597	−	0.999837i	$-0.505749\pi$
$229$	−125.168	−0.0361195	−0.0180597	+	0.999837i	$0.505749\pi$
$230$	308.150	0.0883427
$231$	0	0
$232$	1917.61	0.542660
$233$	5045.95	1.41876	0.709381	−	0.704826i	$-0.248975\pi$
$233$	5045.95	1.41876	0.709381	+	0.704826i	$0.248975\pi$
$234$	36.9176	0.0103136
$235$	−2055.39	−0.570548
$236$	−1572.64	−0.433771
$237$	5043.20	1.38224
$238$	0	0
$239$	−7111.97	−1.92483	−0.962416	−	0.271580i	$-0.912454\pi$
$239$	−7111.97	−1.92483	−0.962416	+	0.271580i	$0.912454\pi$
$240$	−571.163	−0.153618
$241$	−3865.74	−1.03325	−0.516627	−	0.856211i	$-0.672813\pi$
$241$	−3865.74	−1.03325	−0.516627	+	0.856211i	$0.672813\pi$
$242$	−2466.39	−0.655146
$243$	391.559	0.103368
$244$	−3752.54	−0.984557
$245$	0	0
$246$	−3968.28	−1.02849
$247$	389.224	0.100266
$248$	1633.61	0.418285
$249$	−7572.20	−1.92718
$250$	−2748.22	−0.695252
$251$	−4429.65	−1.11393	−0.556966	−	0.830535i	$-0.688035\pi$
$251$	−4429.65	−1.11393	−0.556966	+	0.830535i	$0.688035\pi$
$252$	0	0
$253$	−227.464	−0.0565239
$254$	−4963.80	−1.22621
$255$	2617.41	0.642779
$256$	256.000	0.0625000
$257$	−7644.68	−1.85549	−0.927747	−	0.373209i	$-0.878258\pi$
$257$	−7644.68	−1.85549	−0.927747	+	0.373209i	$0.878258\pi$
$258$	3902.71	0.941754
$259$	0	0
$260$	354.093	0.0844613
$261$	334.826	0.0794070
$262$	3242.12	0.764500
$263$	−4269.52	−1.00103	−0.500513	−	0.865729i	$-0.666855\pi$
$263$	−4269.52	−1.00103	−0.500513	+	0.865729i	$0.666855\pi$
$264$	421.609	0.0982888
$265$	1889.44	0.437989
$266$	0	0
$267$	−2614.76	−0.599329
$268$	−1698.12	−0.387049
$269$	4731.36	1.07240	0.536201	−	0.844090i	$-0.319859\pi$
$269$	4731.36	1.07240	0.536201	+	0.844090i	$0.319859\pi$
$270$	1827.95	0.412019
$271$	−7314.45	−1.63956	−0.819781	−	0.572677i	$-0.805905\pi$
$271$	−7314.45	−1.63956	−0.819781	+	0.572677i	$0.805905\pi$
$272$	−1173.15	−0.261516
$273$	0	0
$274$	−1319.90	−0.291016
$275$	792.410	0.173760
$276$	−490.256	−0.106920
$277$	2661.72	0.577355	0.288677	−	0.957426i	$-0.406785\pi$
$277$	2661.72	0.577355	0.288677	+	0.957426i	$0.406785\pi$
$278$	1211.80	0.261434
$279$	285.239	0.0612073
$280$	0	0
$281$	3721.81	0.790122	0.395061	−	0.918655i	$-0.370723\pi$
$281$	3721.81	0.790122	0.395061	+	0.918655i	$0.370723\pi$
$282$	3270.05	0.690527
$283$	1705.34	0.358204	0.179102	−	0.983830i	$-0.442681\pi$
$283$	1705.34	0.358204	0.179102	+	0.983830i	$0.442681\pi$
$284$	−1554.72	−0.324844
$285$	−1051.44	−0.218534
$286$	−261.377	−0.0540404
$287$	0	0
$288$	44.6992	0.00914558
$289$	463.054	0.0942507
$290$	3211.47	0.650290
$291$	−5907.83	−1.19011
$292$	2316.03	0.464162
$293$	5176.18	1.03207	0.516034	−	0.856568i	$-0.327408\pi$
$293$	5176.18	1.03207	0.516034	+	0.856568i	$0.327408\pi$
$294$	0	0
$295$	−2633.74	−0.519805
$296$	−3335.59	−0.654990
$297$	−1349.31	−0.263620
$298$	−978.569	−0.190225
$299$	303.935	0.0587860
$300$	1707.89	0.328684
$301$	0	0
$302$	−3894.75	−0.742112
$303$	6068.71	1.15062
$304$	471.266	0.0889111
$305$	−6284.49	−1.17983
$306$	−204.839	−0.0382675
$307$	−6788.05	−1.26194	−0.630968	−	0.775809i	$-0.717342\pi$
$307$	−6788.05	−1.26194	−0.630968	+	0.775809i	$0.717342\pi$
$308$	0	0
$309$	−2106.96	−0.387898
$310$	2735.86	0.501247
$311$	−2937.67	−0.535626	−0.267813	−	0.963471i	$-0.586301\pi$
$311$	−2937.67	−0.535626	−0.267813	+	0.963471i	$0.586301\pi$
$312$	−563.350	−0.102222
$313$	10091.1	1.82231	0.911155	−	0.412063i	$-0.135192\pi$
$313$	10091.1	1.82231	0.911155	+	0.412063i	$0.135192\pi$
$314$	−3983.58	−0.715944
$315$	0	0
$316$	−3785.57	−0.673908
$317$	3426.21	0.607051	0.303525	−	0.952823i	$-0.401836\pi$
$317$	3426.21	0.607051	0.303525	+	0.952823i	$0.401836\pi$
$318$	−3006.02	−0.530093
$319$	−2370.58	−0.416072
$320$	428.731	0.0748962
$321$	−6644.79	−1.15538
$322$	0	0
$323$	−2159.62	−0.372027
$324$	−3059.06	−0.524529
$325$	−1058.81	−0.180715
$326$	147.577	0.0250722
$327$	−2332.43	−0.394446
$328$	2978.70	0.501437
$329$	0	0
$330$	706.081	0.117783
$331$	1725.74	0.286571	0.143286	−	0.989681i	$-0.454233\pi$
$331$	1725.74	0.286571	0.143286	+	0.989681i	$0.454233\pi$
$332$	5683.91	0.939594
$333$	−582.415	−0.0958442
$334$	−2283.02	−0.374016
$335$	−2843.89	−0.463816
$336$	0	0
$337$	11169.7	1.80549	0.902746	−	0.430174i	$-0.141548\pi$
$337$	11169.7	1.80549	0.902746	+	0.430174i	$0.141548\pi$
$338$	−4044.75	−0.650904
$339$	−9626.02	−1.54222
$340$	−1964.70	−0.313385
$341$	−2019.50	−0.320710
$342$	82.2861	0.0130103
$343$	0	0
$344$	−2929.49	−0.459149
$345$	−821.046	−0.128127
$346$	−5544.48	−0.861482
$347$	4524.22	0.699921	0.349961	−	0.936764i	$-0.386195\pi$
$347$	4524.22	0.699921	0.349961	+	0.936764i	$0.386195\pi$
$348$	−5109.34	−0.787038
$349$	875.472	0.134278	0.0671389	−	0.997744i	$-0.478613\pi$
$349$	875.472	0.134278	0.0671389	+	0.997744i	$0.478613\pi$
$350$	0	0
$351$	1802.94	0.274171
$352$	−316.472	−0.0479204
$353$	4107.69	0.619350	0.309675	−	0.950842i	$-0.399780\pi$
$353$	4107.69	0.619350	0.309675	+	0.950842i	$0.399780\pi$
$354$	4190.19	0.629113
$355$	−2603.74	−0.389273
$356$	1962.71	0.292201
$357$	0	0
$358$	−5226.71	−0.771621
$359$	5121.96	0.752999	0.376499	−	0.926417i	$-0.377128\pi$
$359$	5121.96	0.752999	0.376499	+	0.926417i	$0.377128\pi$
$360$	74.8592	0.0109595
$361$	−5991.45	−0.873517
$362$	−2544.01	−0.369365
$363$	6571.52	0.950181
$364$	0	0
$365$	3878.72	0.556224
$366$	9998.40	1.42794
$367$	4441.47	0.631724	0.315862	−	0.948805i	$-0.397706\pi$
$367$	4441.47	0.631724	0.315862	+	0.948805i	$0.397706\pi$
$368$	368.000	0.0521286
$369$	520.100	0.0733749
$370$	−5586.20	−0.784900
$371$	0	0
$372$	−4352.66	−0.606653
$373$	9168.19	1.27268	0.636342	−	0.771407i	$-0.280447\pi$
$373$	9168.19	1.27268	0.636342	+	0.771407i	$0.280447\pi$
$374$	1450.26	0.200511
$375$	7322.47	1.00835
$376$	−2454.59	−0.336664
$377$	3167.54	0.432724
$378$	0	0
$379$	−419.664	−0.0568778	−0.0284389	−	0.999596i	$-0.509054\pi$
$379$	−419.664	−0.0568778	−0.0284389	+	0.999596i	$0.509054\pi$
$380$	789.244	0.106546
$381$	13225.7	1.77841
$382$	−4547.52	−0.609087
$383$	−8635.25	−1.15206	−0.576032	−	0.817427i	$-0.695400\pi$
$383$	−8635.25	−1.15206	−0.576032	+	0.817427i	$0.695400\pi$
$384$	−682.095	−0.0906459
$385$	0	0
$386$	−6519.51	−0.859674
$387$	−511.507	−0.0671870
$388$	4434.58	0.580237
$389$	−6256.78	−0.815505	−0.407753	−	0.913092i	$-0.633687\pi$
$389$	−6256.78	−0.815505	−0.407753	+	0.913092i	$0.633687\pi$
$390$	−943.459	−0.122497
$391$	−1686.40	−0.218120
$392$	0	0
$393$	−8638.42	−1.10878
$394$	−10376.7	−1.32683
$395$	−6339.81	−0.807571
$396$	−55.2580	−0.00701217
$397$	−5254.55	−0.664278	−0.332139	−	0.943230i	$-0.607770\pi$
$397$	−5254.55	−0.664278	−0.332139	+	0.943230i	$0.607770\pi$
$398$	2914.28	0.367034
$399$	0	0
$400$	−1281.99	−0.160249
$401$	−8738.31	−1.08821	−0.544103	−	0.839018i	$-0.683130\pi$
$401$	−8738.31	−1.08821	−0.544103	+	0.839018i	$0.683130\pi$
$402$	4524.53	0.561351
$403$	2698.44	0.333545
$404$	−4555.34	−0.560982
$405$	−5123.09	−0.628564
$406$	0	0
$407$	4123.51	0.502198
$408$	3125.77	0.379286
$409$	3467.93	0.419262	0.209631	−	0.977781i	$-0.432774\pi$
$409$	3467.93	0.419262	0.209631	+	0.977781i	$0.432774\pi$
$410$	4988.52	0.600891
$411$	3516.80	0.422070
$412$	1581.54	0.189119
$413$	0	0
$414$	64.2552	0.00762794
$415$	9519.01	1.12595
$416$	422.866	0.0498383
$417$	−3228.75	−0.379167
$418$	−582.588	−0.0681706
$419$	−15477.9	−1.80465	−0.902323	−	0.431061i	$-0.858139\pi$
$419$	−15477.9	−1.80465	−0.902323	+	0.431061i	$0.858139\pi$
$420$	0	0
$421$	−13191.0	−1.52705	−0.763527	−	0.645776i	$-0.776534\pi$
$421$	−13191.0	−1.52705	−0.763527	+	0.645776i	$0.776534\pi$
$422$	4021.49	0.463894
$423$	−428.587	−0.0492639
$424$	2256.41	0.258445
$425$	5874.85	0.670523
$426$	4142.45	0.471132
$427$	0	0
$428$	4987.77	0.563301
$429$	696.423	0.0783767
$430$	−4906.10	−0.550217
$431$	−11868.8	−1.32645	−0.663226	−	0.748419i	$-0.730813\pi$
$431$	−11868.8	−1.32645	−0.663226	+	0.748419i	$0.730813\pi$
$432$	2182.97	0.243121
$433$	−4665.32	−0.517785	−0.258893	−	0.965906i	$-0.583358\pi$
$433$	−4665.32	−0.517785	−0.258893	+	0.965906i	$0.583358\pi$
$434$	0	0
$435$	−8556.76	−0.943138
$436$	1750.79	0.192311
$437$	677.445	0.0741570
$438$	−6170.91	−0.673191
$439$	−12477.6	−1.35654	−0.678270	−	0.734812i	$-0.737270\pi$
$439$	−12477.6	−1.35654	−0.678270	+	0.734812i	$0.737270\pi$
$440$	−530.005	−0.0574249
$441$	0	0
$442$	−1937.83	−0.208536
$443$	−2021.34	−0.216788	−0.108394	−	0.994108i	$-0.534571\pi$
$443$	−2021.34	−0.216788	−0.108394	+	0.994108i	$0.534571\pi$
$444$	8887.45	0.949954
$445$	3287.01	0.350156
$446$	1443.12	0.153214
$447$	2607.33	0.275889
$448$	0	0
$449$	4370.38	0.459356	0.229678	−	0.973267i	$-0.426233\pi$
$449$	4370.38	0.459356	0.229678	+	0.973267i	$0.426233\pi$
$450$	−223.844	−0.0234491
$451$	−3682.32	−0.384465
$452$	7225.56	0.751907
$453$	10377.3	1.07631
$454$	−4670.66	−0.482830
$455$	0	0
$456$	−1255.66	−0.128951
$457$	5733.14	0.586838	0.293419	−	0.955984i	$-0.405207\pi$
$457$	5733.14	0.586838	0.293419	+	0.955984i	$0.405207\pi$
$458$	−250.337	−0.0255403
$459$	−10003.7	−1.01728
$460$	616.301	0.0624677
$461$	3614.55	0.365177	0.182588	−	0.983189i	$-0.441552\pi$
$461$	3614.55	0.365177	0.182588	+	0.983189i	$0.441552\pi$
$462$	0	0
$463$	−9279.48	−0.931434	−0.465717	−	0.884934i	$-0.654204\pi$
$463$	−9279.48	−0.931434	−0.465717	+	0.884934i	$0.654204\pi$
$464$	3835.21	0.383718
$465$	−7289.52	−0.726975
$466$	10091.9	1.00322
$467$	−7961.64	−0.788909	−0.394455	−	0.918915i	$-0.629066\pi$
$467$	−7961.64	−0.788909	−0.394455	+	0.918915i	$0.629066\pi$
$468$	73.8352	0.00729280
$469$	0	0
$470$	−4110.78	−0.403438
$471$	10614.0	1.03836
$472$	−3145.27	−0.306722
$473$	3621.48	0.352042
$474$	10086.4	0.977393
$475$	−2360.00	−0.227967
$476$	0	0
$477$	393.983	0.0378181
$478$	−14223.9	−1.36106
$479$	−10120.4	−0.965368	−0.482684	−	0.875795i	$-0.660338\pi$
$479$	−10120.4	−0.965368	−0.482684	+	0.875795i	$0.660338\pi$
$480$	−1142.33	−0.108625
$481$	−5509.79	−0.522297
$482$	−7731.48	−0.730621
$483$	0	0
$484$	−4932.77	−0.463258
$485$	7426.73	0.695321
$486$	783.118	0.0730925
$487$	−14066.3	−1.30884	−0.654419	−	0.756132i	$-0.727087\pi$
$487$	−14066.3	−1.30884	−0.654419	+	0.756132i	$0.727087\pi$
$488$	−7505.08	−0.696187
$489$	−393.210	−0.0363631
$490$	0	0
$491$	−2489.53	−0.228821	−0.114410	−	0.993434i	$-0.536498\pi$
$491$	−2489.53	−0.228821	−0.114410	+	0.993434i	$0.536498\pi$
$492$	−7936.56	−0.727251
$493$	−17575.2	−1.60558
$494$	778.448	0.0708988
$495$	−92.5422	−0.00840295
$496$	3267.23	0.295772
$497$	0	0
$498$	−15144.4	−1.36273
$499$	19174.5	1.72018	0.860089	−	0.510144i	$-0.170408\pi$
$499$	19174.5	1.72018	0.860089	+	0.510144i	$0.170408\pi$
$500$	−5496.45	−0.491617
$501$	6082.97	0.542449
$502$	−8859.30	−0.787669
$503$	−648.963	−0.0575265	−0.0287633	−	0.999586i	$-0.509157\pi$
$503$	−648.963	−0.0575265	−0.0287633	+	0.999586i	$0.509157\pi$
$504$	0	0
$505$	−7628.97	−0.672247
$506$	−454.928	−0.0399684
$507$	10777.0	0.944028
$508$	−9927.60	−0.867059
$509$	19322.6	1.68263	0.841314	−	0.540547i	$-0.181783\pi$
$509$	19322.6	1.68263	0.841314	+	0.540547i	$0.181783\pi$
$510$	5234.82	0.454513
$511$	0	0
$512$	512.000	0.0441942
$513$	4018.60	0.345859
$514$	−15289.4	−1.31203
$515$	2648.65	0.226628
$516$	7805.43	0.665920
$517$	3034.41	0.258130
$518$	0	0
$519$	14772.9	1.24944
$520$	708.187	0.0597232
$521$	1453.09	0.122190	0.0610949	−	0.998132i	$-0.480541\pi$
$521$	1453.09	0.122190	0.0610949	+	0.998132i	$0.480541\pi$
$522$	669.653	0.0561492
$523$	−18332.1	−1.53271	−0.766356	−	0.642416i	$-0.777932\pi$
$523$	−18332.1	−1.53271	−0.766356	+	0.642416i	$0.777932\pi$
$524$	6484.24	0.540583
$525$	0	0
$526$	−8539.04	−0.707832
$527$	−14972.4	−1.23759
$528$	843.218	0.0695007
$529$	529.000	0.0434783
$530$	3778.87	0.309705
$531$	−549.185	−0.0448825
$532$	0	0
$533$	4920.28	0.399852
$534$	−5229.52	−0.423789
$535$	8353.17	0.675026
$536$	−3396.24	−0.273685
$537$	13926.2	1.11911
$538$	9462.72	0.758302
$539$	0	0
$540$	3655.89	0.291342
$541$	8385.46	0.666394	0.333197	−	0.942857i	$-0.391873\pi$
$541$	8385.46	0.666394	0.333197	+	0.942857i	$0.391873\pi$
$542$	−14628.9	−1.15935
$543$	6778.34	0.535703
$544$	−2346.29	−0.184920
$545$	2932.10	0.230454
$546$	0	0
$547$	−3200.32	−0.250157	−0.125079	−	0.992147i	$-0.539918\pi$
$547$	−3200.32	−0.250157	−0.125079	+	0.992147i	$0.539918\pi$
$548$	−2639.81	−0.205779
$549$	−1310.44	−0.101873
$550$	1584.82	0.122867
$551$	7060.18	0.545869
$552$	−980.512	−0.0756039
$553$	0	0
$554$	5323.44	0.408251
$555$	14884.1	1.13837
$556$	2423.59	0.184862
$557$	12067.9	0.918011	0.459005	−	0.888433i	$-0.348206\pi$
$557$	12067.9	0.918011	0.459005	+	0.888433i	$0.348206\pi$
$558$	570.479	0.0432801
$559$	−4838.99	−0.366132
$560$	0	0
$561$	−3864.13	−0.290809
$562$	7443.61	0.558701
$563$	14391.3	1.07730	0.538651	−	0.842529i	$-0.318934\pi$
$563$	14391.3	1.07730	0.538651	+	0.842529i	$0.318934\pi$
$564$	6540.10	0.488276
$565$	12100.9	0.901039
$566$	3410.68	0.253289
$567$	0	0
$568$	−3109.44	−0.229699
$569$	10147.2	0.747613	0.373806	−	0.927507i	$-0.378052\pi$
$569$	10147.2	0.747613	0.373806	+	0.927507i	$0.378052\pi$
$570$	−2102.89	−0.154527
$571$	11056.1	0.810300	0.405150	−	0.914250i	$-0.367219\pi$
$571$	11056.1	0.810300	0.405150	+	0.914250i	$0.367219\pi$
$572$	−522.755	−0.0382124
$573$	12116.6	0.883380
$574$	0	0
$575$	−1842.86	−0.133657
$576$	89.3985	0.00646690
$577$	−11079.2	−0.799366	−0.399683	−	0.916653i	$-0.630880\pi$
$577$	−11079.2	−0.799366	−0.399683	+	0.916653i	$0.630880\pi$
$578$	926.107	0.0666453
$579$	17370.8	1.24681
$580$	6422.94	0.459825
$581$	0	0
$582$	−11815.7	−0.841538
$583$	−2789.41	−0.198157
$584$	4632.06	0.328212
$585$	123.654	0.00873925
$586$	10352.4	0.729782
$587$	18879.1	1.32747	0.663733	−	0.747970i	$-0.268971\pi$
$587$	18879.1	1.32747	0.663733	+	0.747970i	$0.268971\pi$
$588$	0	0
$589$	6014.59	0.420758
$590$	−5267.48	−0.367557
$591$	27648.1	1.92435
$592$	−6671.17	−0.463148
$593$	−10355.3	−0.717098	−0.358549	−	0.933511i	$-0.616728\pi$
$593$	−10355.3	−0.717098	−0.358549	+	0.933511i	$0.616728\pi$
$594$	−2698.63	−0.186408
$595$	0	0
$596$	−1957.14	−0.134509
$597$	−7764.90	−0.532322
$598$	607.870	0.0415680
$599$	−20615.2	−1.40620	−0.703099	−	0.711092i	$-0.748201\pi$
$599$	−20615.2	−1.40620	−0.703099	+	0.711092i	$0.748201\pi$
$600$	3415.78	0.232415
$601$	−27765.0	−1.88445	−0.942227	−	0.334974i	$-0.891272\pi$
$601$	−27765.0	−1.88445	−0.942227	+	0.334974i	$0.891272\pi$
$602$	0	0
$603$	−593.005	−0.0400482
$604$	−7789.50	−0.524752
$605$	−8261.06	−0.555140
$606$	12137.4	0.813612
$607$	27363.1	1.82971	0.914854	−	0.403784i	$-0.132305\pi$
$607$	27363.1	1.82971	0.914854	+	0.403784i	$0.132305\pi$
$608$	942.533	0.0628697
$609$	0	0
$610$	−12569.0	−0.834268
$611$	−4054.55	−0.268460
$612$	−409.677	−0.0270592
$613$	−11183.1	−0.736834	−0.368417	−	0.929661i	$-0.620100\pi$
$613$	−11183.1	−0.736834	−0.368417	+	0.929661i	$0.620100\pi$
$614$	−13576.1	−0.892324
$615$	−13291.6	−0.871494
$616$	0	0
$617$	22190.9	1.44793	0.723963	−	0.689839i	$-0.242319\pi$
$617$	22190.9	1.44793	0.723963	+	0.689839i	$0.242319\pi$
$618$	−4213.91	−0.274285
$619$	−3415.66	−0.221788	−0.110894	−	0.993832i	$-0.535371\pi$
$619$	−3415.66	−0.221788	−0.110894	+	0.993832i	$0.535371\pi$
$620$	5471.72	0.354435
$621$	3138.02	0.202777
$622$	−5875.33	−0.378745
$623$	0	0
$624$	−1126.70	−0.0722822
$625$	810.494	0.0518716
$626$	20182.2	1.28857
$627$	1552.27	0.0988701
$628$	−7967.16	−0.506249
$629$	30571.3	1.93793
$630$	0	0
$631$	18787.6	1.18530	0.592648	−	0.805461i	$-0.298082\pi$
$631$	18787.6	1.18530	0.592648	+	0.805461i	$0.298082\pi$
$632$	−7571.14	−0.476525
$633$	−10715.0	−0.672801
$634$	6852.42	0.429250
$635$	−16626.1	−1.03903
$636$	−6012.05	−0.374832
$637$	0	0
$638$	−4741.16	−0.294207
$639$	−542.928	−0.0336117
$640$	857.462	0.0529596
$641$	23874.5	1.47112	0.735558	−	0.677461i	$-0.236920\pi$
$641$	23874.5	1.47112	0.735558	+	0.677461i	$0.236920\pi$
$642$	−13289.6	−0.816975
$643$	9322.40	0.571757	0.285878	−	0.958266i	$-0.407715\pi$
$643$	9322.40	0.571757	0.285878	+	0.958266i	$0.407715\pi$
$644$	0	0
$645$	13072.0	0.797998
$646$	−4319.25	−0.263063
$647$	−12616.8	−0.766642	−0.383321	−	0.923615i	$-0.625220\pi$
$647$	−12616.8	−0.766642	−0.383321	+	0.923615i	$0.625220\pi$
$648$	−6118.11	−0.370898
$649$	3888.24	0.235172
$650$	−2117.62	−0.127785
$651$	0	0
$652$	295.155	0.0177288
$653$	−7286.61	−0.436672	−0.218336	−	0.975874i	$-0.570063\pi$
$653$	−7286.61	−0.436672	−0.218336	+	0.975874i	$0.570063\pi$
$654$	−4664.86	−0.278915
$655$	10859.4	0.647802
$656$	5957.40	0.354569
$657$	808.787	0.0480271
$658$	0	0
$659$	−12927.3	−0.764150	−0.382075	−	0.924131i	$-0.624790\pi$
$659$	−12927.3	−0.764150	−0.382075	+	0.924131i	$0.624790\pi$
$660$	1412.16	0.0832854
$661$	−2743.70	−0.161449	−0.0807244	−	0.996736i	$-0.525723\pi$
$661$	−2743.70	−0.161449	−0.0807244	+	0.996736i	$0.525723\pi$
$662$	3451.48	0.202637
$663$	5163.21	0.302447
$664$	11367.8	0.664393
$665$	0	0
$666$	−1164.83	−0.0677721
$667$	5513.12	0.320043
$668$	−4566.05	−0.264470
$669$	−3845.09	−0.222212
$670$	−5687.78	−0.327968
$671$	9277.92	0.533785
$672$	0	0
$673$	−29779.0	−1.70564	−0.852820	−	0.522205i	$-0.825110\pi$
$673$	−29779.0	−1.70564	−0.852820	+	0.522205i	$0.825110\pi$
$674$	22339.3	1.27668
$675$	−10931.9	−0.623359
$676$	−8089.50	−0.460258
$677$	22937.6	1.30216	0.651081	−	0.759008i	$-0.274316\pi$
$677$	22937.6	1.30216	0.651081	+	0.759008i	$0.274316\pi$
$678$	−19252.0	−1.09052
$679$	0	0
$680$	−3929.40	−0.221597
$681$	12444.7	0.700265
$682$	−4039.00	−0.226776
$683$	−2224.80	−0.124640	−0.0623202	−	0.998056i	$-0.519850\pi$
$683$	−2224.80	−0.124640	−0.0623202	+	0.998056i	$0.519850\pi$
$684$	164.572	0.00919968
$685$	−4420.97	−0.246593
$686$	0	0
$687$	667.006	0.0370420
$688$	−5858.98	−0.324668
$689$	3727.18	0.206087
$690$	−1642.09	−0.0905991
$691$	25465.8	1.40198	0.700988	−	0.713173i	$-0.252742\pi$
$691$	25465.8	1.40198	0.700988	+	0.713173i	$0.252742\pi$
$692$	−11089.0	−0.609160
$693$	0	0
$694$	9048.43	0.494919
$695$	4058.87	0.221527
$696$	−10218.7	−0.556520
$697$	−27300.4	−1.48361
$698$	1750.94	0.0949487
$699$	−26889.2	−1.45500
$700$	0	0
$701$	−16844.7	−0.907581	−0.453791	−	0.891108i	$-0.649929\pi$
$701$	−16844.7	−0.907581	−0.453791	+	0.891108i	$0.649929\pi$
$702$	3605.88	0.193868
$703$	−12280.9	−0.658864
$704$	−632.943	−0.0338849
$705$	10952.9	0.585120
$706$	8215.39	0.437946
$707$	0	0
$708$	8380.38	0.444850
$709$	6076.32	0.321863	0.160932	−	0.986966i	$-0.448550\pi$
$709$	6076.32	0.321863	0.160932	+	0.986966i	$0.448550\pi$
$710$	−5207.47	−0.275258
$711$	−1321.97	−0.0697296
$712$	3925.43	0.206617
$713$	4696.64	0.246691
$714$	0	0
$715$	−875.473	−0.0457914
$716$	−10453.4	−0.545618
$717$	37898.7	1.97399
$718$	10243.9	0.532450
$719$	−23769.1	−1.23288	−0.616438	−	0.787403i	$-0.711425\pi$
$719$	−23769.1	−1.23288	−0.616438	+	0.787403i	$0.711425\pi$
$720$	149.718	0.00774954
$721$	0	0
$722$	−11982.9	−0.617670
$723$	20600.0	1.05964
$724$	−5088.02	−0.261180
$725$	−19205.9	−0.983847
$726$	13143.0	0.671879
$727$	15689.1	0.800381	0.400190	−	0.916432i	$-0.368944\pi$
$727$	15689.1	0.800381	0.400190	+	0.916432i	$0.368944\pi$
$728$	0	0
$729$	18562.1	0.943050
$730$	7757.45	0.393310
$731$	26849.3	1.35849
$732$	19996.8	1.00970
$733$	10814.2	0.544928	0.272464	−	0.962166i	$-0.412161\pi$
$733$	10814.2	0.544928	0.272464	+	0.962166i	$0.412161\pi$
$734$	8882.94	0.446697
$735$	0	0
$736$	736.000	0.0368605
$737$	4198.49	0.209842
$738$	1040.20	0.0518839
$739$	223.806	0.0111405	0.00557024	−	0.999984i	$-0.498227\pi$
$739$	223.806	0.0111405	0.00557024	+	0.999984i	$0.498227\pi$
$740$	−11172.4	−0.555008
$741$	−2074.12	−0.102827
$742$	0	0
$743$	14170.6	0.699688	0.349844	−	0.936808i	$-0.386235\pi$
$743$	14170.6	0.699688	0.349844	+	0.936808i	$0.386235\pi$
$744$	−8705.31	−0.428968
$745$	−3277.68	−0.161188
$746$	18336.4	0.899923
$747$	1984.89	0.0972202
$748$	2900.53	0.141783
$749$	0	0
$750$	14644.9	0.713010
$751$	5964.24	0.289798	0.144899	−	0.989446i	$-0.453714\pi$
$751$	5964.24	0.289798	0.144899	+	0.989446i	$0.453714\pi$
$752$	−4909.18	−0.238058
$753$	23605.0	1.14238
$754$	6335.09	0.305982
$755$	−13045.3	−0.628831
$756$	0	0
$757$	25138.8	1.20698	0.603490	−	0.797370i	$-0.293776\pi$
$757$	25138.8	1.20698	0.603490	+	0.797370i	$0.293776\pi$
$758$	−839.328	−0.0402187
$759$	1212.13	0.0579676
$760$	1578.49	0.0753392
$761$	34711.2	1.65346	0.826728	−	0.562601i	$-0.190199\pi$
$761$	34711.2	1.65346	0.826728	+	0.562601i	$0.190199\pi$
$762$	26451.5	1.25753
$763$	0	0
$764$	−9095.03	−0.430689
$765$	−686.099	−0.0324261
$766$	−17270.5	−0.814632
$767$	−5195.43	−0.244584
$768$	−1364.19	−0.0640963
$769$	36718.2	1.72184	0.860918	−	0.508744i	$-0.169890\pi$
$769$	36718.2	1.72184	0.860918	+	0.508744i	$0.169890\pi$
$770$	0	0
$771$	40737.5	1.90289
$772$	−13039.0	−0.607881
$773$	25383.2	1.18107	0.590536	−	0.807011i	$-0.298916\pi$
$773$	25383.2	1.18107	0.590536	+	0.807011i	$0.298916\pi$
$774$	−1023.01	−0.0475084
$775$	−16361.6	−0.758354
$776$	8869.17	0.410289
$777$	0	0
$778$	−12513.6	−0.576649
$779$	10966.9	0.504403
$780$	−1886.92	−0.0866186
$781$	3843.94	0.176117
$782$	−3372.79	−0.154234
$783$	32703.8	1.49264
$784$	0	0
$785$	−13342.8	−0.606658
$786$	−17276.8	−0.784026
$787$	29565.1	1.33911	0.669557	−	0.742761i	$-0.266484\pi$
$787$	29565.1	1.33911	0.669557	+	0.742761i	$0.266484\pi$
$788$	−20753.5	−0.938213
$789$	22751.7	1.02659
$790$	−12679.6	−0.571039
$791$	0	0
$792$	−110.516	−0.00495835
$793$	−12397.1	−0.555148
$794$	−10509.1	−0.469715
$795$	−10068.6	−0.449176
$796$	5828.55	0.259532
$797$	28774.6	1.27886	0.639428	−	0.768851i	$-0.279171\pi$
$797$	28774.6	1.27886	0.639428	+	0.768851i	$0.279171\pi$
$798$	0	0
$799$	22496.8	0.996095
$800$	−2563.98	−0.113313
$801$	685.405	0.0302342
$802$	−17476.6	−0.769478
$803$	−5726.23	−0.251649
$804$	9049.06	0.396935
$805$	0	0
$806$	5396.88	0.235852
$807$	−25212.8	−1.09979
$808$	−9110.69	−0.396674
$809$	21598.9	0.938660	0.469330	−	0.883023i	$-0.344495\pi$
$809$	21598.9	0.938660	0.469330	+	0.883023i	$0.344495\pi$
$810$	−10246.2	−0.444462
$811$	5808.33	0.251490	0.125745	−	0.992063i	$-0.459868\pi$
$811$	5808.33	0.251490	0.125745	+	0.992063i	$0.459868\pi$
$812$	0	0
$813$	38977.8	1.68144
$814$	8247.02	0.355108
$815$	494.304	0.0212451
$816$	6251.54	0.268196
$817$	−10785.7	−0.461865
$818$	6935.86	0.296463
$819$	0	0
$820$	9977.04	0.424894
$821$	−8886.00	−0.377739	−0.188869	−	0.982002i	$-0.560482\pi$
$821$	−8886.00	−0.377739	−0.188869	+	0.982002i	$0.560482\pi$
$822$	7033.60	0.298449
$823$	34957.8	1.48062	0.740312	−	0.672264i	$-0.234678\pi$
$823$	34957.8	1.48062	0.740312	+	0.672264i	$0.234678\pi$
$824$	3163.08	0.133727
$825$	−4222.65	−0.178199
$826$	0	0
$827$	18908.6	0.795062	0.397531	−	0.917589i	$-0.369867\pi$
$827$	18908.6	0.795062	0.397531	+	0.917589i	$0.369867\pi$
$828$	128.510	0.00539377
$829$	−6825.53	−0.285960	−0.142980	−	0.989726i	$-0.545668\pi$
$829$	−6825.53	−0.285960	−0.142980	+	0.989726i	$0.545668\pi$
$830$	19038.0	0.796168
$831$	−14184.0	−0.592101
$832$	845.733	0.0352410
$833$	0	0
$834$	−6457.51	−0.268112
$835$	−7646.89	−0.316924
$836$	−1165.18	−0.0482039
$837$	27860.4	1.15053
$838$	−30955.9	−1.27608
$839$	−47053.9	−1.93621	−0.968106	−	0.250543i	$-0.919391\pi$
$839$	−47053.9	−1.93621	−0.968106	+	0.250543i	$0.919391\pi$
$840$	0	0
$841$	33067.5	1.35583
$842$	−26382.0	−1.07979
$843$	−19833.0	−0.810303
$844$	8042.98	0.328022
$845$	−13547.7	−0.551546
$846$	−857.175	−0.0348348
$847$	0	0
$848$	4512.81	0.182748
$849$	−9087.53	−0.367354
$850$	11749.7	0.474131
$851$	−9589.81	−0.386292
$852$	8284.90	0.333141
$853$	−24509.5	−0.983810	−0.491905	−	0.870649i	$-0.663699\pi$
$853$	−24509.5	−0.983810	−0.491905	+	0.870649i	$0.663699\pi$
$854$	0	0
$855$	275.614	0.0110243
$856$	9975.54	0.398314
$857$	−4904.91	−0.195506	−0.0977529	−	0.995211i	$-0.531165\pi$
$857$	−4904.91	−0.195506	−0.0977529	+	0.995211i	$0.531165\pi$
$858$	1392.85	0.0554207
$859$	−41550.9	−1.65040	−0.825202	−	0.564838i	$-0.808939\pi$
$859$	−41550.9	−1.65040	−0.825202	+	0.564838i	$0.808939\pi$
$860$	−9812.20	−0.389062
$861$	0	0
$862$	−23737.6	−0.937944
$863$	−4023.40	−0.158700	−0.0793501	−	0.996847i	$-0.525284\pi$
$863$	−4023.40	−0.158700	−0.0793501	+	0.996847i	$0.525284\pi$
$864$	4365.95	0.171913
$865$	−18571.0	−0.729980
$866$	−9330.64	−0.366129
$867$	−2467.55	−0.0966580
$868$	0	0
$869$	9359.58	0.365365
$870$	−17113.5	−0.666899
$871$	−5609.98	−0.218240
$872$	3501.58	0.135984
$873$	1548.61	0.0600374
$874$	1354.89	0.0524369
$875$	0	0
$876$	−12341.8	−0.476018
$877$	−19824.6	−0.763317	−0.381659	−	0.924303i	$-0.624647\pi$
$877$	−19824.6	−0.763317	−0.381659	+	0.924303i	$0.624647\pi$
$878$	−24955.1	−0.959219
$879$	−27583.2	−1.05843
$880$	−1060.01	−0.0406056
$881$	25971.7	0.993200	0.496600	−	0.867979i	$-0.334582\pi$
$881$	25971.7	0.993200	0.496600	+	0.867979i	$0.334582\pi$
$882$	0	0
$883$	33396.8	1.27281	0.636405	−	0.771355i	$-0.280421\pi$
$883$	33396.8	1.27281	0.636405	+	0.771355i	$0.280421\pi$
$884$	−3875.65	−0.147457
$885$	14034.9	0.533081
$886$	−4042.69	−0.153292
$887$	760.454	0.0287864	0.0143932	−	0.999896i	$-0.495418\pi$
$887$	760.454	0.0287864	0.0143932	+	0.999896i	$0.495418\pi$
$888$	17774.9	0.671719
$889$	0	0
$890$	6574.03	0.247598
$891$	7563.31	0.284378
$892$	2886.24	0.108339
$893$	−9037.23	−0.338656
$894$	5214.66	0.195083
$895$	−17506.7	−0.653836
$896$	0	0
$897$	−1619.63	−0.0602875
$898$	8740.75	0.324814
$899$	48947.3	1.81589
$900$	−447.688	−0.0165810
$901$	−20680.4	−0.764666
$902$	−7364.65	−0.271858
$903$	0	0
$904$	14451.1	0.531678
$905$	−8521.05	−0.312983
$906$	20754.6	0.761067
$907$	−11224.2	−0.410908	−0.205454	−	0.978667i	$-0.565867\pi$
$907$	−11224.2	−0.410908	−0.205454	+	0.978667i	$0.565867\pi$
$908$	−9341.31	−0.341412
$909$	−1590.78	−0.0580451
$910$	0	0
$911$	12752.3	0.463778	0.231889	−	0.972742i	$-0.425509\pi$
$911$	12752.3	0.463778	0.231889	+	0.972742i	$0.425509\pi$
$912$	−2511.32	−0.0911820
$913$	−14053.1	−0.509408
$914$	11466.3	0.414957
$915$	33489.3	1.20997
$916$	−500.673	−0.0180597
$917$	0	0
$918$	−20007.4	−0.719327
$919$	3343.46	0.120012	0.0600058	−	0.998198i	$-0.480888\pi$
$919$	3343.46	0.120012	0.0600058	+	0.998198i	$0.480888\pi$
$920$	1232.60	0.0441714
$921$	36172.6	1.29417
$922$	7229.10	0.258219
$923$	−5136.24	−0.183165
$924$	0	0
$925$	33407.8	1.18750
$926$	−18559.0	−0.658623
$927$	552.294	0.0195682
$928$	7670.42	0.271330
$929$	−18180.7	−0.642078	−0.321039	−	0.947066i	$-0.604032\pi$
$929$	−18180.7	−0.642078	−0.321039	+	0.947066i	$0.604032\pi$
$930$	−14579.0	−0.514049
$931$	0	0
$932$	20183.8	0.709381
$933$	15654.4	0.549307
$934$	−15923.3	−0.557843
$935$	4857.60	0.169904
$936$	147.670	0.00515679
$937$	4103.46	0.143067	0.0715336	−	0.997438i	$-0.477211\pi$
$937$	4103.46	0.143067	0.0715336	+	0.997438i	$0.477211\pi$
$938$	0	0
$939$	−53774.2	−1.86886
$940$	−8221.55	−0.285274
$941$	−12273.6	−0.425194	−0.212597	−	0.977140i	$-0.568192\pi$
$941$	−12273.6	−0.425194	−0.212597	+	0.977140i	$0.568192\pi$
$942$	21228.0	0.734230
$943$	8563.77	0.295731
$944$	−6290.55	−0.216885
$945$	0	0
$946$	7242.97	0.248932
$947$	−24222.4	−0.831176	−0.415588	−	0.909553i	$-0.636424\pi$
$947$	−24222.4	−0.831176	−0.415588	+	0.909553i	$0.636424\pi$
$948$	20172.8	0.691121
$949$	7651.34	0.261721
$950$	−4720.00	−0.161197
$951$	−18257.8	−0.622556
$952$	0	0
$953$	32057.0	1.08964	0.544821	−	0.838552i	$-0.316597\pi$
$953$	32057.0	1.08964	0.544821	+	0.838552i	$0.316597\pi$
$954$	787.966	0.0267415
$955$	−15231.7	−0.516112
$956$	−28447.9	−0.962416
$957$	12632.5	0.426699
$958$	−20240.7	−0.682618
$959$	0	0
$960$	−2284.65	−0.0768091
$961$	11907.3	0.399696
$962$	−11019.6	−0.369320
$963$	1741.79	0.0582850
$964$	−15463.0	−0.516627
$965$	−21836.8	−0.728448
$966$	0	0
$967$	−7061.63	−0.234836	−0.117418	−	0.993083i	$-0.537462\pi$
$967$	−7061.63	−0.234836	−0.117418	+	0.993083i	$0.537462\pi$
$968$	−9865.54	−0.327573
$969$	11508.4	0.381529
$970$	14853.5	0.491666
$971$	−22006.0	−0.727298	−0.363649	−	0.931536i	$-0.618469\pi$
$971$	−22006.0	−0.727298	−0.363649	+	0.931536i	$0.618469\pi$
$972$	1566.24	0.0516842
$973$	0	0
$974$	−28132.6	−0.925489
$975$	5642.26	0.185330
$976$	−15010.2	−0.492278
$977$	14370.1	0.470562	0.235281	−	0.971927i	$-0.424399\pi$
$977$	14370.1	0.470562	0.235281	+	0.971927i	$0.424399\pi$
$978$	−786.420	−0.0257126
$979$	−4852.68	−0.158419
$980$	0	0
$981$	611.398	0.0198985
$982$	−4979.06	−0.161801
$983$	−23459.4	−0.761180	−0.380590	−	0.924744i	$-0.624279\pi$
$983$	−23459.4	−0.761180	−0.380590	+	0.924744i	$0.624279\pi$
$984$	−15873.1	−0.514244
$985$	−34756.5	−1.12430
$986$	−35150.5	−1.13531
$987$	0	0
$988$	1556.90	0.0501331
$989$	−8422.28	−0.270792
$990$	−185.084	−0.00594179
$991$	−18287.2	−0.586189	−0.293094	−	0.956084i	$-0.594685\pi$
$991$	−18287.2	−0.586189	−0.293094	+	0.956084i	$0.594685\pi$
$992$	6534.45	0.209142
$993$	−9196.24	−0.293891
$994$	0	0
$995$	9761.25	0.311008
$996$	−30288.8	−0.963592
$997$	38741.1	1.23063	0.615317	−	0.788280i	$-0.289028\pi$
$997$	38741.1	1.23063	0.615317	+	0.788280i	$0.289028\pi$
$998$	38349.0	1.21635
$999$	−56886.7	−1.80162

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.n.1.1	✓	6
7.6	odd	2	inner	2254.4.a.n.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2254.4.a.n.1.1	✓	6	1.1	even	1	trivial
2254.4.a.n.1.6	yes	6	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -5.32887 q^{3} +4.00000 q^{4} +6.69892 q^{5} -10.6577 q^{6} +8.00000 q^{8} +1.39685 q^{9} +13.3978 q^{10} -9.88974 q^{11} -21.3155 q^{12} +13.2146 q^{13} -35.6977 q^{15} +16.0000 q^{16} -73.3216 q^{17} +2.79370 q^{18} +29.4542 q^{19} +26.7957 q^{20} -19.7795 q^{22} +23.0000 q^{23} -42.6310 q^{24} -80.1245 q^{25} +26.4291 q^{26} +136.436 q^{27} +239.701 q^{29} -71.3953 q^{30} +204.202 q^{31} +32.0000 q^{32} +52.7011 q^{33} -146.643 q^{34} +5.58741 q^{36} -416.948 q^{37} +58.9083 q^{38} -70.4187 q^{39} +53.5914 q^{40} +372.338 q^{41} -366.186 q^{43} -39.5590 q^{44} +9.35739 q^{45} +46.0000 q^{46} -306.824 q^{47} -85.2619 q^{48} -160.249 q^{50} +390.721 q^{51} +52.8583 q^{52} +282.051 q^{53} +272.872 q^{54} -66.2506 q^{55} -156.957 q^{57} +479.402 q^{58} -393.159 q^{59} -142.791 q^{60} -938.136 q^{61} +408.403 q^{62} +64.0000 q^{64} +88.5233 q^{65} +105.402 q^{66} -424.530 q^{67} -293.286 q^{68} -122.564 q^{69} -388.680 q^{71} +11.1748 q^{72} +579.008 q^{73} -833.896 q^{74} +426.973 q^{75} +117.817 q^{76} -140.837 q^{78} -946.393 q^{79} +107.183 q^{80} -764.764 q^{81} +744.676 q^{82} +1420.98 q^{83} -491.175 q^{85} -732.372 q^{86} -1277.33 q^{87} -79.1179 q^{88} +490.678 q^{89} +18.7148 q^{90} +92.0000 q^{92} -1088.16 q^{93} -613.648 q^{94} +197.311 q^{95} -170.524 q^{96} +1108.65 q^{97} -13.8145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 24 q^{4} + 48 q^{8} - 52 q^{9} - 84 q^{11} + 52 q^{15} + 96 q^{16} - 104 q^{18} - 168 q^{22} + 138 q^{23} - 286 q^{25} - 22 q^{29} + 104 q^{30} + 192 q^{32} - 208 q^{36} - 180 q^{37} - 414 q^{39}+ \cdots + 600 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 24 * q^4 + 48 * q^8 - 52 * q^9 - 84 * q^11 + 52 * q^15 + 96 * q^16 - 104 * q^18 - 168 * q^22 + 138 * q^23 - 286 * q^25 - 22 * q^29 + 104 * q^30 + 192 * q^32 - 208 * q^36 - 180 * q^37 - 414 * q^39 - 600 * q^43 - 336 * q^44 + 276 * q^46 - 572 * q^50 + 464 * q^51 - 876 * q^53 - 1452 * q^57 - 44 * q^58 + 208 * q^60 + 384 * q^64 - 1532 * q^65 - 92 * q^67 - 1434 * q^71 - 416 * q^72 - 360 * q^74 - 828 * q^78 - 2820 * q^79 - 2198 * q^81 - 184 * q^85 - 1200 * q^86 - 672 * q^88 + 552 * q^92 - 2742 * q^93 - 2888 * q^95 + 600 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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